Question 787:

Asked on July 30, 2009

Tags: correlation coefficient , significant correlation , Pearson r

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1. Q787_1_correlation_q787.xls

I've done the computation as well as used Megastat for this question. Megastat doesn't tell you the degrees of freedom or givens you a p-value so I've provided that as well.

A correlation tells you the degree of relationship between two quantitative variables.

Part a. In the attached excel file I've included the scatter plot of the fertility rates for 1990 and 2000 from both megastat and just using excel.

Part b. The correlation coefficient can be found using excel's CORREL() or PEARSON() functions. I also computed it by hand so you can see that it's basically just the sum of the product of the deviation scores. The correlation coefficient r is .749.  This shows a strong positive association between the fertility rates between 1990 and 2000 (meaning a high rate in 1990 is indicative of a high rate in 2000). 

Part c. To test the significance of a correlation coefficient, you use the t-distribution on n-2 degrees of freedom and use the following formula  t = r / sqrt[(1—r2)/(N—2)].

r is the correlation coefficient, r2 is the r-squared (.561) and n is the sample size. Filling in the values we get

t = .749 / sqrt[(1—.561)/(15—2)] = 4.0799.

Looking up the t critical value of 4.0799 in a t-table or using the excel function =TDIST() on n-2 = 13 degrees of freedom we get the p-value of .001.

Since the p-value is low (less than .01) we'd conclude that the correlation is statistically significant.